

for all multi indices \(\alpha \in \mathbb {N}^2\)  satisfying \(|\alpha | \le \operatorname{deg}(\Sigma )+1\) , where \(v^\alpha \)  are as defined in Lemma REF . The local equations (REF ) are equivalent to (REF ). Indeed, (REF )-() imply \((\operatorname{Div}\Sigma )|_{\Omega -O}=0\)  or, in other words, \(\operatorname{Div}\Sigma  \in \mathcal {E}(\Omega ,\mathbb {R}^2)\) . Equation (), on using Lemma REF , then implies (REF ). Similarly, we use the divergence identities [1]}, in addition to Identity REF , to deduce the local form of (REF ):
\(\operatorname{div}\operatorname{div}m - \langle \sigma , \lambda  \rangle =f_1 ~\text{in}~\Omega -\lbrace S\cup \lbrace O\rbrace \rbrace , \\\begin{aligned}\left\langle \llbracket \operatorname{div}m \rrbracket , \nu  \right\rangle +\left\langle \frac{d\llbracket m\rrbracket }{ds} ,t\otimes \nu  \right\rangle -\left\langle \frac{d^2n}{ds^2},t\otimes t \right\rangle - k^2\langle n,t\otimes t \rangle -k\langle \llbracket m\rrbracket ,t\otimes t \rangle \\ + \left\langle \lbrace \sigma \rbrace , \nu \otimes \nu  \right\rangle \left\langle \gamma , \nu \otimes \nu  \right\rangle + \left\langle \lbrace \lambda \rbrace , t\otimes t \right\rangle \left\langle \tau , t\otimes t \right\rangle =f_2 ~\text{on}~S-\lbrace O\rbrace ,\end{aligned} \\\left\langle \frac{dn}{ds},t\otimes \nu \right\rangle - k \left\langle n,t\otimes t \right\rangle - \langle \llbracket m\rrbracket ,\nu \otimes \nu  \rangle =0 ~\text{on}~S-\lbrace O\rbrace , \\\left\langle n , \nu \otimes \nu  \right\rangle =0 ~\text{on}~S-\lbrace O\rbrace ,~\text{and} \\M(\nabla \nabla v^\alpha )+\nabla \Phi \otimes \nabla W (\mathbb {A} \nabla \nabla v^\alpha )=F(v^\alpha )~\text{at}~O,\) 
